It looks like something has stat()ed /opt/bin/ici and then decided
it's been asked to execute that, and refusing (which makes a kind
of sense), and bailing out with an error (*that* step seems wrong
to me).

Well, didn't you ask to execute '/opt/bin/ici'? After all, that's
what you typed. I don't see how it could be wrong to report back
what you asked to execute is a directory.

I thought that bash treated the first word on the line (after
optional assignments to environment variables a la FRED=x
run-some-command) as a command to execute, passing the remaining
words on the line to the command as arguments? (Leaving aside things
like backtic execution and variable expansion.)

So I still think I asked for /opt/bin/ici to be executed by bash.
I'd be interested to know if I've misunderstood.

I think you did as well. And so does bash. But it's not going to allow
you to execute a directory, which is what your invocation matches
exactly.
So it tells you that you made a mistake by trying to execute a directory.
I guess we're in violent agreement that you got what you asked for.

Well, not violent, but clear disagreement. :-)
I think that given I can type "foo" to execute foo or foo.exe, why does
having a directory called foo make bash decide that I want to do the
impossible: namely, execute the directory? Why is it written to assume
that the user is trying to do something that makes no sense?

I suspect it won't become clear until I have time to grab the bash
source and probably the exec source and read them through myself.

I also checked that bash doesn't work this way under Linux. I
created a directory called ici (with execute permission, obviously),
in the first directory in my PATH. I then ran ici from bash, and it
did not tell me that ici was a directory and bail out - it executed
the first ici executable it found later in my PATH.

Well here you're not comparing the same thing at all. You can't put a
file/binary named "ici" in the same spot as a directory you have named
"ici". So you've already changed the rules. But try the exact same
thing you just did under Linux on Cygwin and you'll see the same
behavior as Linux.

I had to move the Ici directory aside, but I confirmed that you're
correct. So it looks like you understand what's going on better than I
do, since I don't understand why and you do. :-)

The point is that Windows allows you to do something
you can't do in Linux. You can have the name of an executable match the
name of a directory, if you ignore the extension. In addition you can
run an executable by only providing part of its name (i.e. not the
extension). You can't do this under Linux. And why would you want to.

Quite!

But if you try to put that same-named executable and directory under the
one directory and then run the executable from there without providing
the full name, you're being imprecise. Cygwin's bash lets you know this.
You can't compare this behavior to Linux because you'd never get into
that situation.

True, but given that Cygwin is designed to allow the user to be
imprecise (drop the .exe suffix) when asking for a command to be
executed, why have that logic (designed to improve usability because of
the "interesting" decision to identify executables by suffix) decide
that if the name matches a directory, then the user is really trying to
execute a directory. One thing we agree on, is that it makes no sense
to try to execute a directory.

Don't confuse any of this with an executable named "ici.exe" in one
directory in your path and a directory named "ici" in another (also
in your path). This isn't a general issue that will bite you every
time you want to run "ici.exe" in this configuration. In this scenario,
the only time running "ici.exe" as "ici" would cause you to get the
complaint that "ici" is a directory is if you were trying to run it from
the parent directory of "ici"-the-directory.

Well, I'm getting the problem regardless of what directory I try to run
the command from. I don't get the problem only if I'm trying to run ici
when my current directory is /opt/bin.

Unless you mean that the problem only occurs when I specify the full
pathname, and that pathname is a directory of that name and there is a
.exe file there based on the same name, too.

Which would make sense because in that case the search down PATH is not
done, where it looks for a command with the right name (maybe with .exe
or .lnk extension) to run...

Which makes me think that exec() (or execve or whatever is the exec()
that searches down the PATH) is the place where the check for .exe with
rejection of directories, makes sense. That change would fix the
problem in all shells, too, not just bash.

And I'm sorry if you are now tearing your hair out wondering why I don't
get it,still. :-(

And if you take that parent
directory (and "." if you have that) out of your path, you can run
"ici.exe"
as "ici" from anywhere.

Better?

I'm afraid I may still be misunderstanding, sorry. Do you mean, move
the ici directory to some place away from where the ici.exe lives?